Jacqui Lambie hopes for return to Canberra

Outgoing senator Jacqui Lambie believes she has unfinished business in federal politics.

Jacqui Lambie believes she has unfinished business in federal politics after becoming the latest MP to quit parliament over the citizenship fiasco.

The colourful Tasmanian senator made a tearful goodbye to the upper house on Tuesday as she confirmed her UK citizenship through her Scottish-born father.

"There's so much more I wanted to do here and I hope to get another chance to keep going at it," Senator Lambie told parliament.

"I don't really care which side it's on, I do hope to come back."

She hasn't ruled out returning to Canberra via the lower house if Labor MP Justine Keay faces a by-election in her Tasmanian seat of Braddon.

The Turnbull government is continuing to pressure the opposition to refer Ms Keay and her party room colleague Susan Lamb to the High Court over their citizenship doubts.

As one chapter closed on Senator Lambie's political career, former NSW premier Kristina Keneally reopened the book on hers.

Tuesday's shock announcement that she will run for Labor in the Bennelong by-election injected intrigue to the contest in John Howard's former seat.

The opposition is hopeful Ms Keneally can apply pressure to Liberal MP John Alexander, whose dual citizenship triggered the December 16 by-election.

More immediately, the government will deal with the fallout from the national marriage survey results which will be released by the Australian Bureau of Statistics at 10am (AEDT) on Wednesday.

The Senate will be asked on Wednesday to support a move to introduce draft laws allowing same-sex marriage on Thursday.

Liberal senator Dean Smith has lodged a notice of motion for his private senator's bill to be brought on, one day after the result of the national survey is released.

Meanwhile, the High Court will consider whether Liberal Hollie Hughes should be declared duly elected to replace cabinet minister Fiona Nash in the Senate.

Ms Hughes' appointment is under question due to her job at the Administrative Appeals Tribunal, a position which as been argued is an "office of profit under the crown" - a reason for disqualification set out in the constitution.